To determine the dimensions of the cube box that will fit the spherical soap, we first need to find the radius of the sphere from the given surface area. The formula for the surface area \( A \) of a sphere is:
\[ A = 4\pi r^2 \]
Where:
- \( A \) is the surface area
- \( r \) is the radius
Given that the surface area of the soap is 63.585 in² and using 3.14 as the value of π, we can set up the equation:
\[ 63.585 = 4 \times 3.14 \times r^2 \]
Calculating \( 4 \times 3.14 \):
\[ 4 \times 3.14 = 12.56 \]
Now, plugging this back into the equation:
\[ 63.585 = 12.56 \times r^2 \]
Next, we solve for \( r^2 \):
\[ r^2 = \frac{63.585}{12.56} \approx 5.06 \]
Now, take the square root to find \( r \):
\[ r \approx \sqrt{5.06} \approx 2.25 \text{ in.} \]
The diameter \( d \) of the sphere, which is twice the radius, is:
\[ d = 2r \approx 2 \times 2.25 = 4.5 \text{ in.} \]
Since Marcus wants to package the spherical soap in a cube box snugly, the side length of the cube box should equal the diameter of the sphere.
Thus, the dimensions of the cube box should be:
\[ \text{4.5 in.} \times \text{4.5 in.} \times \text{4.5 in.} \]
Therefore, the correct answer is:
4.5 in. × 4.5 in. × 4.5 in.
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